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Questions or comments concerning this laboratory should be directedto Prof. Charles A. Bouman, School of Electrical and Computer Engineering, Purdue University, West Lafayette IN 47907;(765) 494-0340; bouman@ecn.purdue.edu

Bivariate distributions

In this section, we will study the concept of a bivariate distribution.We will see that bivariate distributions characterize how two random variables are related to each other.We will also see that correlation and covariance are two simple measures of the dependencies between random variables,which can be very useful for analyzing both random variables and random processes.

Background on bivariate distributions

Sometimes we need to account for not just one random variable, but several. In this section, we will examine the case of two randomvariables–the so called bivariate case–but the theory is easily generalized to accommodate more than two.

The random variables X and Y have cumulative distribution functions (CDFs) F X ( x ) and F Y ( y ) , also known as marginal CDFs. Since there may be an interaction between X and Y , the marginal statistics may not fully describe their behavior.Therefore we define a bivariate , or joint CDF as

F X , Y ( x , y ) = P ( X x , Y y ) .

If the joint CDF is sufficiently “smooth”, we can define a joint probability density function,

f X , Y ( x , y ) = 2 x y F X , Y ( x , y ) .

Conversely, the joint probability density function may be used to calculate thejoint CDF:

F X , Y ( x , y ) = - y - x f X , Y ( s , t ) d s d t .

The random variables X and Y are said to be independent if and only if their joint CDF (or PDF) is a separable function, which means

f X , Y ( x , y ) = f X ( x ) f Y ( y )

Informally, independence between random variables means that one random variable does not tell you anything about the other.As a consequence of the definition, if X and Y are independent, then the product of their expectations is the expectation of their product.

E [ X Y ] = E [ X ] E [ Y ]

While the joint distribution contains all the information about X and Y , it can be very complex and is often difficult to calculate. In many applications, a simple measure of the dependenciesof X and Y can be very useful. Three such measures are the correlation , covariance , and the correlation coefficient .

  • Correlation
    E [ X Y ] = - - x y f X , Y ( x , y ) d x d y
  • Covariance
    E [ ( X - μ X ) ( Y - μ Y ) ] = - - ( x - μ X ) ( y - μ Y ) f X , Y ( x , y ) d x d y
  • Correlation coefficient
    ρ X Y = E [ ( X - μ X ) ( Y - μ Y ) ] σ X σ Y = E [ X Y ] - μ X μ Y σ X σ Y

If the correlation coefficient is 0, then X and Y are said to be uncorrelated . Notice that independence implies uncorrelatedness,however the converse is not true.

Samples of two random variables

In the following experiment, we will examine the relationship between the scatter plots for pairs of random samples ( X i , Z i ) and their correlation coefficient. We will see that the correlation coefficient determines the shape ofthe scatter plot.

Let X and Y be independent Gaussian random variables, each with mean 0 and variance 1. We will consider the correlation between X and Z , where Z is equal to the following:

  1. Z = Y
  2. Z = ( X + Y ) / 2
  3. Z = ( 4 * X + Y ) / 5
  4. Z = ( 99 * X + Y ) / 100

Notice that since Z is a linear combination of two Gaussian random variables, Z will also be Gaussian.

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
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